prompt listlengths 1 1 | label stringlengths 1 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知复数列 $\\left\\{z_{n}\\right\\}$ 满足: $$ z_{1}=\\frac{\\sqrt{3}}{2}, z_{n+1}=\\overline{z_{n}}\\left(1+z_{n} \\mathrm{i}\\rig... | 2033 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A, B$ 为平面直角坐标系 $x O y$ 中的曲线 $x y=1(x, y>0)$ 上的两点, 向量 $\\vec{m}=(1,|O A|)$ ，则数量积 $\\vec{m} \\cdot \\overrightarrow{O B}$ 的最小... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $\\cos A, \\sin A, \\tan B$ 依次构成以 $\\frac{3}{4}$ 为公比的等比数列,则 $\\cot C$ 的原始答案是以$-\\frac{m}{n}$的形式给出，请给出$... | 149 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中，椭圆 $\\Gamma: \\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1$ ，点 $P$ 在椭圆 $\\Gamma$ 内部，且在直线 $y=x$ 上移动。点 $K, L$ 在 $\\Gamma$ 上，满足 $... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知非负实数 $a, b, c$ 满足 $a+b+c=1$, 则 $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$ 的最大值为 $\\frac{m+\\sqrt{n}}{p}$的格式，请给出 $m + n + p$ 的值。\n\nR... | 12 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n定义 $x * y=\\frac{x+y}{1+x y}$, 则 $(\\cdots((2 * 3) * 4) \\cdots) * 21$结果的格式为$\\frac{m}{n}$，给出$m + n$的值。\n\nRemember to put you... | 231 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若非负数 $a, b, c$ 满足 $a+b+c=1$ ，则 $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$ 的最大值是 $\\frac{\\sqrt{m}}{n}$ 的格式，请给出 $m + n$ 的值。\n\nRemember... | 21 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知矩形 $A B C D$ 中, $A B=2, B C=1$, 折叠使点 $A, C$ 重合, 折痕为 $M N$, 打开平面 $A D M N$, 使二面角 $A-M N-C$ 为 $\\frac{\\pi}{3}$, 求直线 $M N$ 与直线... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知在 $\\triangle A B C$ 中, $D$ 是边 $B C$ 的中点, 且 $\\angle D A C=15^{\\circ}$, 求 $\\angle A B C$ 的最大值。原始答案为 $\\frac{m\\pi}{n}$ 的形式... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正方体 $A B C D-A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime}$ 的棱长为 1, 点 $M, N$ 分别为线段 $A B^{\\prime}, A C$ 上的动点, 点 $T$在屏幕 $B ... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $\\mathrm{I}_{\\mathrm{n}}=\\left(1+\\frac{\\mathrm{i}}{2}\\right)\\left(1+\\frac{\\mathrm{i}}{4}\\right)\\left(1+\\frac{\\... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求椭圆 $C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ 的内接六边形面积的最大值。已知原始答案的格式为$\\frac{m\\sqrt{3}}{n}$，请给出$m + n$的值。\n\nRemember ... | 5 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 $\\left\\{a_{n}\\right\\}$ ，且 $S=n a+n_{n}(n-1)$，点 $\\left(a_{n}, \\frac{S_{n}}{n}\\right)$ 所在的直线方程的斜率的形式为$\\frac{n}{m}$，... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $\\operatorname{Re}(z) \\geq 1$, 求 $\\left|\\frac{1}{z}+1+\\mathrm{i}\\right|_{\\text {min }}$ 的值。原始答案为 $\\frac{\\sqrt{m}-n... | 16 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $x^{2}=4 y, M(2,2)$, 过 $M$ 点的直线交抛物线于 $A, B$ 两点, 过 $A, B$ 两点作抛物线的切线交与点 $P$, 求 $S_{\\triangle A B P}$ 的最小值。请给出该最小值。\n\nRememb... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $a x^{3}+b x^{2}+x+1=0(a<0)$ 恰有两个零点, 求 $a+b$ 的取值范围. 请给出区间右端点的整数部分。\n\nRemember to put your answer on its own line after \"A... | 0 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $\\sin A=\\frac{\\sqrt{2}}{2}$, $\\cos B+\\sqrt{2} \\cos C$ 的取值范围的形式为$(a,b] \\cup(c, \\sqrt{d}]$，求$a+... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在椭圆 $\\Gamma$ 中, $A$ 为长轴的一个端点, $B$ 为短轴的一个端点, $F_{1}, F_{2}$ 为两个焦点. 若 $\\overrightarrow{A F_{1}} \\cdot \\overrightarrow{A F_{2... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a, b \\in \\mathbf{R}$ 。若关于 $z$ 的方程\\[\\left(z^{2}+a z+b\\right)\\left(z^{2}+a z+2 b\\right)=0\\]有 4 个互不相等的复数根 $z_{1}, z_{2... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n平面向量 $\\boldsymbol{a}, \\boldsymbol{b}$ 夹角为 $\\frac{\\pi}{3}$, 若 $|\\boldsymbol{a}|,|\\boldsymbol{b}|,|\\boldsymbol{a}+\\bolds... | 15 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 $f(x)$ 的定义域为 $\\mathbf{R}$, 且当 $x \\geq 0$ 时, $f(x)=|x-2|+a$ (其中 $a$ 为实数). 若 $f(x)$ 为奇函数, 则不等式 $f(x) \\geq 1$ 的解集为$[a,b] \... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正整数数列 $\\left\\{a_{n}\\right\\}$ 同时具有以下两个性质：\\n（i）对任意正整数 $k$ ，均有 $a_{2 k-1}+a_{2 k}=2^{k}$ ；\\n(ii) 对任意正整数 $m$, 均存在正整数 $l \\l... | 1021 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $\\triangle A B C$ 满足 $\\frac{\\angle A}{3}=\\frac{\\angle B}{4}=\\frac{\\angle C}{5}$, 则 $\\frac{\\overrightarrow{A B} \\cd... | 10 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于和为 1 的九个非负实数 $a_{1}, a_{2}, \\cdots, a_{9}$, 令 \n$$ \n\\begin{aligned} \n& S=\\min \\left\\{a_{1}, a_{2}\\right\\}+2 \\mi... | 319 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若四棱雉 $P-A B C D$ 的棱 $A B, B C$ 的长均为 $\\sqrt{2}$, 其他各条棱长均为 1, 则该四棱雉的体积为$\\frac{\\sqrt{m}}{n}$的形式，请给出$m + n$的值。\n\nRemember to p... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若复数 $z$ 满足: $\\frac{z-3 i}{z+i}$ 为负实数 ( i 为虚数单位), $\\frac{z-3}{z+1}$ 为纯虚数, 则 $z$ 的值为$\\sqrt{m} \\mathrm{i}$ 的形式，请给出 $m$ 的值。\n\... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 $f(x)=\\frac{x^{2}+x+16}{x}(2 \\leq x \\leq a)$, 其中实数 $a>2$. 若 $f(x)$ 的值域为 $[9,11]$, 请给出 $a$ 的取值范围端点的和。\n\nRemember to put... | 12 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正数 $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ 满足: $a_{1}, a_{2}, a_{3}$ 成公差为 $b_{1}$ 的等差数列, $b_{1}, b_{2}, b_{3}$ 成公比为 $a_{1}... | 29 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中, $F_{1}, F_{2}$ 是双曲线 $\\Gamma: \\frac{x^{2}}{3}-y^{2}=1$ 的两个焦点, $\\Gamma$ 上一点 $P$ 满足 $\\overrightarrow{P F_{1}} \\cd... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若正四棱雉 $P-A B C D$ 的各条棱长均相等, $M$ 为棱 $A B$ 的中点, 则异面直线 $B P$ 与 $C M$ 所成的角的余弦值为 $\\frac{\\sqrt{m}}{n}$ 的形式，请给出m + n的值。\n\nRemember... | 15 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a=1+10^{-4}$ 。在 $2023 \\times 2023$ 的方格表的每个小方格中填入区间 $[1, a]$ 中的一个实数。设第 $i$ 行的总和为 $x_{i}$ ，第 $i$ 列的总和为 $y_{i}$ ， $1 \\leq i ... | 4046 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n方程 $\\sin x=\\cos 2 x$ 的最小的 $20$ 个正实数解之和为$m\\pi$, 请给出$m$的值。\n\nRemember to put your answer on its own line after \"Answer:\"."... | 130 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $m, n$ 是给定的整数， $m \\geq n \\geq 3$ 。求具有下述性质的最小正整数 $k$ ：若将 $1,2, \\cdots, k$ 中的每个数任意染为红色或者蓝色，则或者存在 $m$ 个红色的数 $x_{1}, x_{2}, \... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求出所有满足下面要求的不小于 -1 的实数 $t$ ：对任意 $a \\in[-2, t]$, 总存在 $b, c \\in[-2, t]$, 使得 $a b+c=1$. 请给出满足条件的最小整数。\n\nRemember to put your an... | -1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $z, w$ 满足 $z+w=2$, 求 $S=\\left|z^{2}-2 w\\right|+\\left|w^{2}-2 z\\right|$的最小可能值。原始答案为$a\\sqrt{b}-c$的格式，请给出$a+b+c$的值。\n\nR... | 29 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, 已知 $\\cos C=\\frac{\\sin A+\\cos A}{2}=\\frac{\\sin B+\\cos B}{2}$,求 $\\cos C$ 的值. 该答案的原始形式为 \\(\\frac... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=\\frac{\\pi}{6}, a_{n+1}=\\arctan \\left(\\sec a_{n}\\right)\\left(n \\in \\mathbb{N}^... | 3333 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在椭圆 $\\Omega$ 中 $F_{1}, F_{2}$ 为焦点, $A$ 为长轴的一个端点, $B$ 为短轴的一个端点, 若 $\\angle F_{1} B F_{2}=\\angle F_{1} A B$ ，则 $\\Omega$ 的离心率为... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n方程 $\\cos \\frac{X}{4}=\\cos x$ 的通解是 $\\qquad$ ，在 $(0,24 \\pi)$ 内不相同的解有多少个？请给出不相同解的个数。\n\nRemember to put your answer on its o... | 20 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n方程 $\\cos \\frac{x}{4}=\\cos x$ 的通解是 $\\qquad$, 在 $(0,24 \\pi)$ 内不相同的解有多少个？请给出在 $(0,24 \\pi)$ 内不相同的解的个数加1的值。\n\nRemember to pu... | 21 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如图， $A B$ 是单位圆的直径，在 $A B$ 上任取一点 $D$ ，作 $D C \\perp A B$, 交圆周于 $C$ 。若点 $D$ 的坐标为 $(x, 0)$,则当 $x \\in$ $\\qquad$时，线段 $A D 、 B D 、... | 0 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如图, 在正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中, $E$ 是 $B C$ 中点, $F$ 在 $A A_{1}$ 上, 且 $A_{1} F: F A=1: 2$. 求平面 $B_{1} E F$与底面 $A_... | 40 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如图，在正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中， $E$ 是 $B C$ 的中点， $F$ 在 $A A_{1}$ 上，且 $A_{1} F: F A=1: 2$, 求平面 $B_{1} E F$ 与底面 $A_... | 40 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n长为 $\\sqrt{2}$ ，宽为 1 的矩形，以它的一条对角线所在的直线为轴旋转一周，求得到的旋转体的体积。原始答案为 \\frac{m}{n} \\sqrt{k \\pi} 的形式，请给出 m + n + k 的值。\n\nRemember to... | 98 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$(\\sqrt{x}+2)^{2 n+1}$ 的展开式中, $x$ 的整数次幂的各项系数之和为 $\\qquad$.请给出\nn=2时的结果\n\nRemember to put your answer on its own line after \... | 122 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知复数数列 $\\left\\{z_{n}\\right\\}$ 满足 $z_{1}=1, z_{n+1}=\\overline{z_{n}}+1+n \\mathrm{i}$ ，其中 $n=1,2, \\cdots$ ，其中 i 是虚数单位, $\... | 2015 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a_{1}, a_{2}, a_{3}, a_{4}$ 是 4 个有理数, 使得\n$$\n\\left\\{a_{i} a_{j} \\mid 1 \\leqslant i<j \\leqslant 4\\right\\}=\\left\\{-... | 0 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的实数 $\\lambda$的八倍， 使得不等式\n$$\n\\sum_{k=1}^{n} x_{k}^{3}\\left(x_{k}-x_{k-1}\\right) \\geq \\frac{1}{4}+\\frac{\\lambda}{n}\... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate$$\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum_{k=1}^{n}\\left(\\left[\\frac{2n}{k}\\right]-2\\left[\\frac{n}{k}\\rig... | -1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite \"word... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remai... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $v$ be a vertex (corner) of a cube $C$ with edges of length 4. Let $S$ be the largest sphere that can be inscribed in $C$.... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that \n$P(u x... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a solid sphere of radius $R$ the density $\\rho$ is a function of $r$, the distance from the center of the sphere. If the m... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\min (x, y)$ denote the smaller of the numbers $x$ and $y$. For what $\\lambda$'s does the equation \n$$\\int_{0}^{1} \... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it starte... | 126 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\alpha and \\beta be given positive real numbers, with \\alpha<\\beta. If two points are selected at random from a straig... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若不等式 $\\sqrt{x}+\\sqrt{y} \\leqslant k \\sqrt{5 x+y}$ 对任意的正实数 $x 、 y$ 都成立, 则实数 $k$ 的最小值为$\\frac{\\sqrt{m}}{n}$的格式，请给出$m + n$的值... | 35 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x_{1}=3, x_{n+1}=\\sqrt{x_{n}+14}-\\sqrt{x_{n}+2}\\left(n \\in N^{*}\\right)$, 求当$n=100$时$\\left[x_{1}+x_{2}+\\cdots+x_{n}\... | 200 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven positive integers $m$ and $n \\geq m$, determine the largest number of dominoes ( $1 \\times 2$ or $2 \\times 1$ rectang... | 27 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the numbe... | 1010 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all pairs of positive real numbers $(x, y)$ such that $x y$ is an integer and\n$$\nx+y=\\left\\lfloor x^{2}-y^{2}\\right\... | 101 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOlja writes down $n$ positive integers $a_{1}, a_{2}, \\ldots, a_{n}$ smaller than $p_{n}$ where $p_{n}$ denotes the $n$-th pr... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. Let $B_{n}$ be the set of all binary strings of length $n$. For a binary string $s_{1} s_{2} \\... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all polynomials $P$ with integer coefficients such that $P(0) \\neq 0$ and \n$$ \nP^{n}(m) \\cdot P^{m}(n) \n$$ \nis ... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知矩阵\n$$\nA=\\left[\\begin{array}{ccc}\n2 & 0 & 0 \\\\\n1 & 3 & 1 \\\\\n-1 & 1 & 3\n\\end{array}\\right], A^{10}=\\left[\\begi... | 524800 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正四面体可容纳 10 个半径为 1 的小球，则正四面体棱长的最小值为？答案的格式为$a+b\\sqrt{c}$，请给出$a+b+c$的值。\n\nRemember to put your answer on its own line after \... | 12 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $\\triangle A B C$ 为椭圆 $\\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1$ 的内接三角形，且 $A B$ 过点 $P(1,0)$ ，则 $\\triangle A B C$ 的面积的最大值为 ? 请给... | 21 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n过抛物线 $E: y^{2}=2 x$ 的焦点 F 作两条斜率之积为 $-\\frac{1}{2}$ 的直线 $l_{1}, l_{2}$, 其中 $l_{1}$ 交 $E$ 于 $A, C$ 两点, $l_{2}$ 交 E 于 $B, D$ 两点, ... | 0 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定奇素数 $p$ 和整数 $a$. 若$p\\equiv-1(mod4)$，且$p|a$时，求方程 $x^{2}+y^{2} \\equiv a(\\bmod p)$ 在模 $p$ 意义下的解的个数。\n\nRemember to put your ... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a sequence $a_{1}<a_{2}<\\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) with $1 \\leq i<$ $j \\leq n$ is called \\un... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA domino is a $2 \\times 1$ or $1 \\times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without o... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe $n$ contestants of an EGMO are named $C_{1}, \\ldots, C_{n}$. After the competition they queue in front of the restaurant ... | 502 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. Dominoes are placed on a $2 n \\times 2 n$ board in such a way that every cell of the board is ... | 21 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the... | 924 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知对一切的 $x \\in \\mathbf{R}$ 恒有 $3 \\sin ^{2} x-\\cos ^{2} x+4 a \\cos x+a^{2} \\leqslant 31$. 求实数 $a$ 的取值范围. 请给出区间的端点和.\n\nRe... | 0 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_{1}, a_{2} \\ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $k$, find the smallest number $n_{k}$ for which there exist real $n_{k} \\times n_{k}$ matrices $A_{... | 12 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the following sequence \\(\\left(a_{n}\\right)_{n=1}^{\\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \\ldots)\\). Find al... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all complex numbers $\\lambda$ for which there exist a positive integer $n$ and a real $n \\times n$ matrix $A$ such... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor $R>1$ let $\\mathcal{D}_{R}=\\left\\{(a, b) \\in \\mathbb{Z}^{2}: 0<a^{2}+b^{2}<R\\right\\}$. Compute \n$$ \n\\lim _{R \... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA four-digit number $Y E A R$ is called very good if the system $$ \\begin{aligned} & Y x+E y+A z+R w=Y \\\\ & R x+Y y+E z... | 12309 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设整数 $a_{1}, a_{2}, \\cdots, a_{2019}$ 满足 $1=a_{1} \\leqslant a_{2} \\leqslant \\cdots \\leqslant a_{2019}=99$. 记 $f=\\left(a_{... | 2016 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n8 个女孩与 25 个男孩围成一圈，任何两个女孩之间至少站两个男孩，则共有多少种不同的排列方法？答案标准形式为$C^{a}_{b} \\cdot c! \\cdot d!$计算出a+b+c+d的值。\n\nRemember to put your a... | 55 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind a necessary and sufficient condition on the positive integer $n$ that the equation\n\n$$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$\n\n... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n8 个女孩与 25 个男孩围成一圈，任何两个女孩之间至少站两个男孩，那么，共有多少种不同的排列方法？标准答案形式为$\\frac{a!\\cdot b!}{c!}$，计算出a+b+c的值。\n\nRemember to put your answer ... | 50 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$n^{2}(n \\geqslant 4)$ 个正数排成 $n$ 行 $n$ 列, $$ \\begin{array}{llllll} a_{11} & a_{12} & a_{13} & a_{14} & \\cdots \\cdots & a_{... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正三棱雉 $P-A B C$ 的高为 $P O, M$ 为 $P O$ 的中点, 过 $A M$ 作与棱 $B C$ 平行的平面, 将三棱雉截为上、下两部分，试求此两部分的体积比。请给出比值中两个数之和的值。\n\nRemember to put y... | 25 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $O$ 为抛物线的顶点, $F$ 为焦点, 且 $P Q$ 为过 $F$ 的弦. 已知 $|O F|=a,|P Q|=$ $b$, 求 $\\triangle O P Q$ 的面积。请提供答案的格式为 a \\sqrt{ab}，当a=b=1时，给出... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $z_{1}, z_{2}$ 都是复数, 且 $\\left|z_{1}\\right|=3,\\left|z_{2}\\right|=5\\left|z_{1}+z_{2}\\right|=7$, 则 $\\arg \\left(\\frac{z... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n二次方程 $(1-i) x^{2}+(\\lambda+i) x+(1+i \\lambda)=0(i$ 为虚数单位, $\\lambda \\in R)$ 有两个虚根的充分必要条件是 ，$\\lambda$ 不可能的的取值为 $\\qquad$.\... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $z \\in C, \\arg \\left(z^{2}-4\\right)=\\frac{5 \\pi}{6}, \\arg \\left(z^{2}+4\\right)=\\frac{\\pi}{3}$, 则 $z$ 的值是 $\\qquad... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知有向线段 $P Q$ 的起点 $P$ 和终点 $Q$ 的坐标分别为 $(-1,1)$ 和 $(2,2)$, 若直线 $l: x+m y+m=0$ 与 $P Q$的延长线相交，则 $m$ 的取值范围是 。请提供区间端点的积。\n\nRemember ... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知一平面与一正方体的 12 条棱的夹角都等于 $\\alpha$, 则 $\\sin \\alpha$ 的原始答案是 \\frac{\\sqrt{m}}{n} 的形式，请给出 m + n 的值。\n\nRemember to put your ans... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$x$ 的二次方程 $x^{2}+z_{1} x+z_{2}+m=0$ 中, $z_{1}, z_{2}, m$ 均是复数，且 $z{ }_{1}^{2}-4 z_{2}=16+20 i$ ，设这个方程的两个根 $\\alpha 、 \\beta$, ... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知复数 $Z_{1} 、 Z_{2}$ 满足 $\\left|Z_{1}\\right|=2,\\left|Z_{2}\\right|=3$, 若它们所对应的向量的夹角为 $60^{\\circ}$, 则 $\\left|\\frac{Z_{1}+Z... | 140 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将八个半径都为 1 的球分放两层放置在一个圆柱内，并使得每个球都和其相邻的四个球相切，且与圆柱的一个底面及侧面都相切。假设原始答案为$m+\\sqrt[n]{j}$的形式，请给出m+n+j的和。\n\nRemember to put your answ... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the length of the curve $y^{2}=x^{3}$ from the origin to the point where the tangent makes an angle of $45^{\\circ}$ with... | 40 |
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